Eurographics Symposium on Geometry Processing 2008Pierre Alliez and Szymon Rusinkiewicz(Guest Editors)

Volume 27 (2008), Number 5

Deformation-Driven Shape Correspondence

H. Zhang1, A. Sheffer2, D. Cohen-Or3, Q. Zhou2, O. van Kaick1, and A. Tagliasacchi1

1Simon Fraser University, 2University of British Columbia, 3Tel Aviv University

Abstract

Non-rigid 3D shape correspondence is a fundamental and difficult problem. Most applications which require a cor-

respondence rely on manually selected markers. Without user assistance, the performances of existing automatic

correspondence methods depend strongly on a good initial shape alignment or shape prior, and they generally do

not tolerate large shape variations. We present an automatic feature correspondence algorithm capable of han-

dling large, non-rigid shape variations, as well as partial matching. This is made possible by leveraging the power

of state-of-the-art mesh deformation techniques and relying on a combinatorial tree traversal for correspondence

search. The search is deformation-driven, prioritized by a self-distortion energy measured on meshes deformed

according to a given correspondence. We demonstrate the ability of our approach to naturally match shapes which

differ in pose, local scale, part decomposition, and geometric detail through numerous examples.

1. Introduction

Establishing a meaningful shape correspondence is a funda-mental task in geometry processing. In applications such asobject recognition, statistical shape modeling, shape morph-ing, and deformation transfer, shape correspondence is oftenthe first step. However, the problem is essentially ill-posedas one generally needs to understand the shape semantics totruly infer what a meaningful correspondence is.

In this paper, we develop an automatic, feature-based cor-respondence algorithm for shapes represented by trianglemeshes. It is aimed at handling large, non-rigid shape vari-ations, as well as partial matching. Removing the rigidityconstraint complicates the problem significantly; see Figure1. Recognizing that the models are both dinosaurs, a humancan easily find a correspondence. However, the large vari-ations in the models’ pose, local scale, and geometric de-tail, e.g., those around the hands, feet, and belly areas, wouldchallenge an automatic algorithm. There is no longer a low-dimensional pose space, as in the rigid case. Also, a highertolerance for local feature dissimilarity and shape distortionmust be allowed, which not only enlarges the solution searchspace but also increases the likelihood of a false match.

Several well-known problems in graphics which requirefeature correspondence, e.g., cross-parameterization, havemostly relied on user input. When dealing with a large modelset, such user assistance may become tedious and impracti-

Figure 1: The dino-skeleton is deformed to match the rap-

tor (red markers indicate features). Top two candidate corre-

spondences are shown. Switching between symmetric parts,

highlighted in circle, is detected by the distortion cost.

cal. The majority of existing approaches for automatic cor-respondence either adopt greedy local search, making themsensitive to initial alignment and prone to local minima, orhandle only rigid or affine transforms. Non-rigid schemeshave been proposed in the context of statistical shape mod-eling. They often benefit from given shape priors, e.g., op-erating only on human faces, bodies, brain surfaces, corpuscallosi shapes, or other medical data. Many works in com-puter vision and medical image analysis are in this category,where the considered shapes do not significantly differ.

To deal with large shape variations, we rely on non-rigidmesh deformation to drive a combinatorial search for thebest correspondence. Our notion of a good feature corre-spondence requires that each corresponding feature pair is

c© 2008 The Author(s)Journal compilation c© 2008 The Eurographics Association and Blackwell Publishing Ltd.Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and350 Main Street, Malden, MA 02148, USA.

H. Zhang & A. Sheffer & D. Cohen-Or & Q. Zhou & O. van Kaick & A. Tagliasacchi / Deformation-Driven Shape Correspondence

sufficiently similar — a local criterion — and that deformingthe shapes into each other while respecting the feature pair-ing incurs a small self-distortion cost — a global criterion. Itis important to note that the cost we use is only measured onthe deformed mesh and not between two meshes. This notonly gives us efficiency but also facilitates partial matching.

We further stipulate that effective shape correspondenceshould be carried out in a coarse-to-fine manner. A smallnumber of features, perhaps up to 10-12, are sufficient torepresent the prominent parts of most objects. A corre-spondence between them can significantly narrow down thesearch space for a more refined mapping, e.g., a dense cross-parameterization, which is a more local problem. The mainchallenge is to find the initial correspondence automatically;this typically involves an expensive global search.

Our algorithm returns a coarse correspondence by operatingon a small number of automatically selected, well-spread,and prominent features on two meshes. The local criterionserves as constraints for a search which is is deformation-

driven, i.e., it is completely prioritized by a distortion en-ergy associated with mesh deformation. Figure 1 highlightsthe utility of the deformation-based approach. Since the twodinosaurs differ rather significantly in their geometric detail,local feature-to-feature similarities become less reliable incorrespondence search. Without any thresholding by featuresimilarity, the deformation-driven search still correctly findsthe most natural correspondence. It also properly differenti-ates symmetry switchings in a shape, such as the crossed legsand arms shown in Figure 1; these are generally difficult todetect by any approach that is completely intrinsic.

The main contributions of our work are:

• A deformation-driven approach to correspondence whichcan handle shapes that vary in pose, local scale, part de-composition, and geometric detail effectively.

– Our distortion cost is intrinsic to the shape and suitableto use for non-rigid correspondence.

– Measuring the distortion on deformed meshes allowsus to detect switching between symmetric parts.

– The deformation-based criterion for correspondence isglobal and can tolerate local shape variations.

– Use of self-distortion cost facilitates partial matchingsince spurious parts do not constrain cost estimates.

• Correspondence search via combinatorial tree traversal toenable partial matching and exploration of a large solutionspace so as to avoid reliance on initial shape alignmentand erroneous local minima reached by greedy search.

With a few initial parameters set, our algorithm runs fullyautomatic feature selection and correspondence. We showthat various models can be matched correctly with fixed pa-rameters and demonstrate the use of the computed matchesas initial correspondences for dense cross-parameterization.

2. Related work

A good example where feature correspondence is neededis cross-parameterization, where a continuous mappingbetween two surfaces is sought. Existing methods, e.g.,[SAPH04, KS04], require a sparse initial correspondencegiven by the user. Although such a correspondence maybe removed eventually [SAPH04], the guidance it pro-vides is indispensable as it immediately brings the start-ing point of the solution search near the global optimumto facilitate gradient descent. In related applications suchas shape morphing [COSL98,GSL∗98,ACOL00], statisticalshape modeling [ACP03, ASK∗05], and deformation trans-fer [SP04,ZRKS05], user assistance is similarly required. Inpractice, it is desirable to minimize user interaction as manu-ally specifying markers is an iterative process, often includ-ing a corrective loop that can be tedious [ZRKS05]. Userinvolvement is especially burdensome when dealing with alarge set of models, i.e., during preparation of training data.

Existing methods for automatic feature correspondencework on salient features extracted based on some local shapesignature [GCO06, GMGP05, LG05]. A proper feature-to-feature similarity provides a means for local feature fitting,followed by a more expensive global search for the top corre-spondences. Our review focuses on correspondence search.

Correspondence search: Branch-and-bound-based combi-natorial search [GMGP05], graph optimization [HH03],voting in pose space [GCO06, LG05], and forwardsearch [HFG∗06], have been proposed for rigid registration.These schemes all benefit from the rigidity constraint, as itimplies a low-dimensional pose space, more reliable featurepicking, and more stringent global consistency thresholdswhich lead to more aggressive pruning in branch-and-bound.Iterated closest point (ICP) [BM92] is applicable in both therigid and non-rigid [CR03] settings, but as a greedy approachICP is sensitive to initial alignment.

Matching of articulated shapes is naturally supported byskeletal [SSGD03] or Reeb graph [TS04, BMSF06] repre-sentations, whose nodes describe geometric features overa shape by a signature. Also, spectral embeddings [EK03]based on intrinsic shape information such as geodesics canprovide pose normalization. However, these structural de-scriptions are expensive to compute accurately. They are alsosensitive to topological changes while being oblivious to ex-trinsic information which should be accounted for in corre-spondence, e.g., a switching between the symmetric parts ofan animal. User assistance is typically required to break suchsymmetries [ASK∗05]. Our deformation-based approach re-lies on intrinsic representations to factor out rigid trans-forms. It also addresses the above issues with the structuraldescriptions as our distortion cost is measured on deformedmeshes, which accounts for extrinsic geometric quantities.

Partial matching: Partial matching allows shapes to corre-spond based only on their sub-parts or sub-regions, offering

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H. Zhang & A. Sheffer & D. Cohen-Or & Q. Zhou & O. van Kaick & A. Tagliasacchi / Deformation-Driven Shape Correspondence

a powerful and versatile option for object recognition andretrieval [GCO06,FS06]. Techniques have been proposed torelax the combinatorial search for partial matchings into acontinuous one, so that constrained non-linear optimizationcan be applied. Relaxation labeling [RHZ76] and graduatedassignments [GR96] fall into this category. Other methodsfor partial matching include geometric hashing [GCO06]and combinatorial searches which rely on upper-bounding k,the number of unmatched features, e.g., [SH81], fixing k inadvance, e.g., [FS06], or heuristically searching for k. Forexample, Gelfand et al. [GMGP05] run their branch-and-bound search of global registrations for each k and chooseone which causes a sharp increase in the minimum regis-tration error returned. The above approaches have all beenapplied to rigid or affine shape registration.

Deformation-based approaches: Early work of Sederbergand Greenwood [SG92] defines a good 2D shape blendingas one which minimizes “work,” combining bending andstretching. Finding an optimal 2D contour correspondence isgreatly simplified by the order preservation constraint (alongthe contours). Blanz and Vetter [BV99] develop morphablemodels for 3D faces and rely on planar parameterizations toreduce the problem to image registration. Sheldon [She00]extends morphable models to surfaces, but the model-fittingterm of the deformation energy is based on Euclidean closestpoints and the energy is minimized via gradient decent. Also,all the test models are already quite similar and roughlyaligned. Much the same can be said about other deformation-based works in computer vision and medical imaging, e.g.,[VG05], where the corresponded shapes, e.g., corpus callo-sum data or human faces, do not possess the kind of largeshape variabilities as the type of models we consider.

3. Overview

The uncanny ability of humans to match complex shapescan be attributed to the initial shape recognition process,which heavily draws upon prior knowledge [Heb49]. Onecan hardly expect a machine to perform recognition ade-quately, which makes automatic correspondence difficult.

3.1. Difficulty of non-rigid shape correspondence

In the case of rigid or affine shape registration, the spaceof allowable transformations is low-dimensional and can becharacterized analytically. One can then define an optimiza-tion criterion accordingly. Although loss of precision can oc-cur in the presence of noise or sampling artifacts, such errorsare relatively predictable and easy to bound. Non-rigid cor-respondence, on the other hand, is an ill-posed problem aswe do not have a definition of what a meaningful matchingis. The amount of shape variations that should be tolerated ismodel-dependent and can take on arbitrary values.

Consider the two features on the belly of the dino-skeletonand the raptor, as shown in Figure 1. Geometrically, their

local neighborhoods are not similar at all, yet semanticallyspeaking, they should correspond. Similar claims can also bemade for the fingers or mouths of the two models. Resort-ing to scale-space theory [Lin94], one may detect a higherdegree of similarity between these features after sufficientsmoothing. However, too much smoothing hinders our abil-ity to differentiate. Indeed, if our non-rigid correspondencealgorithm is to tolerate rather large shape variations, findinga high-quality local shape signature becomes quite difficult,as there is always a conflict between shape discriminationand tolerance of shape variations.

Consequently, our approach is to rely more on global crite-ria, as local shape similarity may become less reliable. Ourdeformation-driven algorithm offers a number of advantagesfor non-rigid shape correspondence (see Section 1).

3.2. Overview of deformation-driven correspondence

Due to correlation between object recognition and shape cor-respondence, as well as the general consensus that our vi-sual object recognition is likely guided by the the objects’part structures [Heb49], we approach the non-rigid corre-spondence problem by considering a coarse set of featuresresiding on the prominent parts of a shape. The features aredetected shape extremities, as described in Section 4.

We define a correspondence or (partial) matching betweenthe features extracted from two meshes as a set of one-to-one

feature pairs. The correspondence search is carried out by acombinatorial tree traversal, which we cover in Section 5.The search is prioritized by a symmetric self-distortion en-ergy (Section 5.1), serving as the correspondence cost. Thedistortion energies are evaluated on deformed meshes withdeformation anchored by the given correspondence. For thistask, we adopt the linear, rotation-invariant differential meshdeformation scheme of Lipman et al. [LSLCO05].

Figure 2 shows an example of our correspondence cost atwork. The correct matching (b), between 9 features on thewolves, has the lowest cost. Leg or ear switches lead tohigher distortion on the deformed meshes, which is capturedby our cost. Note that the same does not hold for the purelyintrinsic geodesic distortion measure β , which accounts forthe total differences between pairwise geodesic distances(see Section 5.2) measured on the original meshes. We reit-erate that a key point of our deformation-driven approach isto measure distortion on the deformed meshes.

To search through the partial matchings between features,we organize them into a search tree. Each path from the rootof the tree encodes a candidate solution to the correspon-dence problem, where each node along the path adds a newfeature pair. We offer the option of early termination whena desired set of solutions has been obtained. To return a sin-gle best solution automatically, we again rely on a distor-tion cost driven approach to choose a matching size (Sec-tion 5.3). To narrow the search space, we employ several

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H. Zhang & A. Sheffer & D. Cohen-Or & Q. Zhou & O. van Kaick & A. Tagliasacchi / Deformation-Driven Shape Correspondence

(a) Two wolves with 10 features (red markers).

(b) β = 0.308. (c) β = 0.330. (d) β = 0.349.

(e) β = 0.348. (f) β = 0.345. (g) β = 0.327.

Figure 2: Top six matchings sorted by distortion cost η: (b)

- (g). The left wolf in (a) is deformed into the other, based

on 9 matched features. Deformation is anchored by extrin-

sic geometry and properly sorts out switches (highlighted in

circle) between the legs or ears. Total geodesic distortion β ,

fails to accomplish that; observe (g) vs. (c) and (e) vs. (d).

pruning techniques, e.g., those based on pairwise geodesicdistances and feature-to-feature similarity defined by curva-ture maps [GGGZ05]; this is covered in Section 5.2.

4. Automatic feature selection for correspondence

Geometric features on a shape can be defined in differ-ent ways, e.g., ridge and valley lines [OBS04], prominenttip points [ZMT05], or points with most unusual signa-tures [GMGP05]. Line-type features are generally not sta-ble under shape articulation. The most prominent featuresof a model part are arguably near its extremity. Extremi-ties are stable under bending and stretching, making themsuitable to use for feature correspondence. Using shape ex-tremities as features also reinforces the correlation betweencorrespondence analysis and object recognition by parts. Ex-tremity features have been utilized for mesh parameteriza-tion [ZMT05] and segmentation [KLT05].

We rely on critical point detection over the average squared

geodesic distance field [HSKK01, ZMT05] defined overa mesh to extract extremities. On each mesh, we find asmall number of uniformly distributed samples s1, . . . ,sk viageodesic farthest point sampling. The geodesic distance fieldG (v) = 1

k ∑ki=1 g(v,si)

2 approximates the average squaredgeodesic distances from vertex v to all other vertices. Thelocal maxima of G correspond to shape extremities (eitherconvex or concave) while the local minima lie near the cen-ter of the shape. We extract the latter as well, since they tend

Figure 3: Features extracted from the T-rex. Left: a coarse

set of 8 features, where σ = 0.05 and γ = 0.2. Right: with

γ = 0.1, additional features (shown in yellow) are selected.

to be stable representatives of the central part of an objectand make the features well-spread.

To keep the feature set small, we smooth G via a polyno-mial, 2t3

σ3 −3t2

σ2 +1, having kernel size σ , where t is geodesicdistance to the center of the smoothing kernel, and applygeodesic Poisson disk sampling with radius γ to the criticalpoints returned. The σ and γ are constrained to [0,1] andmultiplied by the largest pairwise geodesic distance over themesh. Whenever two critical points are found in the samePoisson disk, we throw away the one that is less prominent,i.e., one having a smaller G value. The remaining criticalpoints are sorted by G and selected from atop the sorted list.

We observe that the most prominent extremities can alwaysbe reliably selected. While different choices γ , σ , and featurecount can induce spurious features, the ability of our algo-rithm to perform partial matching ensures that such featureswould be excluded in the result. Figure 3 shows features au-tomatically extracted from a T-rex. With γ to 0.1, we ob-tain a more refined set of features. For coarse feature corre-spondence however, we do not attempt to match up the fine-detail features, e.g., all the individual fingers of the T-rex andthe raptor (see Figure 1). Instead, our correspondence searchonly considers results with a small number of feature pairs.

5. Correspondence search

Let M and M′ be two input meshes with extracted features.We wish to search for the best correspondences between (asubset of) these features in an exponentially large searchspace. Our algorithm traverses a search tree, whose root isthe empty set. Any other node is marked by a feature pair anda path from the root specifies a correspondence. A node (i, i′)can be expanded by a feature pair ( j, j′), yielding a childnode, if the pairs satisfy the pruning constraints and featurej is after feature i according to some arbitrary feature or-dering. This way, all possible (partial) correspondences thatsatisfy the constraints are uniquely represented in the tree.Our correspondence search is based on a best-first strategyand is implemented by a priority queue. At each iteration, thelowest-cost node is extracted from the queue and expanded.The cost of a correspondence is defined by a distortion en-ergy. We seek correspondences which lead to low-distortiondeformations anchored by the corresponding features.

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H. Zhang & A. Sheffer & D. Cohen-Or & Q. Zhou & O. van Kaick & A. Tagliasacchi / Deformation-Driven Shape Correspondence

Figure 4: Importance of symmetric cost. The dino-skeleton

from Figure 1 (7 features) is matched with the T-rex from Fig-

ure 3 (8 features). There is no feature selected on the back

of the dino-skeleton. Blue markers indicate unmatched fea-

tures. Left: optimal correspondence based on a one-sided

cost, i.e., cost of deforming the T-rex into the dino-skeleton

is unaccounted for. We see a mismatch. Right: with a sym-

metric cost, the optimal correspondence is the natural one.

5.1. Mesh deformation and correspondence cost

Given a correspondence π , we deform mesh M into meshM′ using the differential deformation scheme of Lipmanet al. [LSLCO05]. As we compute deformations multipletimes during our tree search, we require a method whichis both efficient and reasonably robust; the differential de-formation scheme addresses both our concerns. The inputto the method consists of anchor points with their associ-ated coordinate frames after the deformation. In our case,the anchor points are the matched feature vertices in π . Theanchors are associated with the corresponded feature vertexnormals, smoothed via polynomial filtering. Given the newanchor positions and normals, the new frames are estimatedusing a rigid transformation mapping the original positionsof the anchors to the new ones.

We use the deformation result to establish a deformation-based cost for the correspondence π . We observe that an er-ror metric based on rotated input normals [LSLCO05] is lessaccurate than a metric using the actual normals after defor-mation, as the accuracy of the former depends on the accu-racy of the normal rotation. Thus we use the error functionaldefined in [KS06] to measure distortion. It uses the deformedmesh to estimate the normals, providing a more accurate er-ror lower bound. We note that the actual deformation methoddescribed in [KS06] is not suitable for our needs as it solvesa non-linear optimization and is fairly slow.

Denote the resulting distortion error by η1(π). We sym-

metrize the cost by also deforming M′ into M, resulting inan error η2(π). The final correspondence cost is η(π) =max{η1(π),η2(π)}. Figure 4 gives an example illustratingthe importance of a symmetric cost measure.

Note that we do not combine distortion energy with featuresimilarity or other terms into the cost as that would requireweighting between different terms. Also, the distortion ismeasured between the deformed mesh M and its original,i.e., a self-distortion, and not between M and M′ since:

1. π is only a sparse feature correspondence. Estimating adistortion between the two meshes M and M′ via a densecorrespondence is non-trivial and expensive; it requiressolving the cross-parameterization problem [KS04].

2. Using a global shape descriptor to compare the deformedand target meshes is not suitable in the presence of spu-rious object parts. The same can be said about the Haus-dorff distance, which is also more expensive to compute.

5.2. Tree pruning

Tree pruning is implemented by enforcing constraints when-ever a node is expanded. A key requirement is that if a nodefails a constraint check then so will all of its descendants.

Local feature similarity: Let (i, i′) be the feature pair defin-ing a new node and denote by s∗ the maximum pairwisefeature dissimilarity. If the dissimilarity distance between i

and i′ exceeds εsims∗, then the node is pruned. Here εsim ∈[0,1] is a tolerance on feature dissimilarity. To compute fea-ture dissimilarity in the non-rigid setting, we use a slightmodification of the curvature map signature of Gatzke etal. [GGGZ05]. For a mesh vertex p, we construct b equallyspaced geodesic rings around p — two adjacent rings bounda geodesic bin. We compute curvature averages in each bin,resulting in a b-dimensional signature. The dissimilarity be-tween two signatures is given by the L2 norm.

Geodesic distortion: Given a new node which extends acorrespondence π , we check whether the new pair incurs ageodesic distortion with some feature pair in π that exceedsεgeod , where εgeod ∈ [0,1] is a tolerance value to control thelevel of tolerable stretching. If this is the case, the new nodeis pruned. We define the geodesic distortion for two featurepairs (i, i′) and ( j, j′) by max{||Gi j −G′

i′ j′ ||, ||G ji −G′j′i′ ||},

where G and G′ are the normalized pairwise geodesic dis-tance matrices for features on M and M′, respectively. Toreduce possible influence of outlier features, we normalizethe original geodesic distances into [0,1] on a per-row basis.As a result, G and G′ are not symmetric in general.

Tolerance values: For non-rigid correspondence, one can-not possibly predict, under all circumstances, the level ofshape variations that should be tolerated to produce a rea-sonable matching. In our current work, we determine thetolerances εsim and εgeod empirically and leave fully auto-matic determination of their values for future work. Experi-mentally, we have found that setting the tolerances conserva-tively allows us to correctly match a wide variety of shapes,due to the effectiveness of the distortion energy.

5.3. Optimal matching size

Each node in the search tree represents a candidate solu-tion: a partial matching between subsets of features from twoinput shapes. All solution candidates of the same size are

c© 2008 The Author(s)Journal compilation c© 2008 The Eurographics Association and Blackwell Publishing Ltd.

H. Zhang & A. Sheffer & D. Cohen-Or & Q. Zhou & O. van Kaick & A. Tagliasacchi / Deformation-Driven Shape Correspondence

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(a) Cost curve. (b) Mismatched wolf.

Figure 5: Finding the right matching size. (a) Plot of low-

est correspondence costs against matching size, for the wolf

pair in Figure 2. (b) Deformed wolf based on 10 feature

pairs. There is a mismatch between a feature on the belly

and a feature on a leg, causing a large jump in the curve.

ordered by their correspondence costs. For automatic cor-respondence, we determine the optimal matching size by afairly standard scheme. We plot the lowest correspondencecosts, denoted by ci, as more features are included, where i

runs from 5 (the smallest matching size we request) to thesize of the largest candidate solution found. We look for apoint i∗ that is the first peak in the difference curve definedby di = ci+1 − ci; this represents the first significant jumpin the correspondence cost. Indeed, we expect a pair of un-matchable features to incur such a jump in distortion.

Figure 5(a) shows the cost curve for the wolf pair from Fig-ure 2. The optimal matching size detected is 9 and the corre-spondence is shown in Figure 2(c). The lowest-cost match-ing of size 10 is shown in Figure 5(b), where we see a pairof mismatched features; they cause a large distortion and ajump in the cost curve. To obtain a meaningful cost curve,we require at least three cost values.

5.4. Implementation issues

While the original input meshes can be dense, we findthat performing mesh deformation and measuring costs onreduced models to be sufficient; this can greatly reducesearch time. In our implementation, we decimate a high-resolution input mesh down to having between 2,500 and5,000 vertices. Note that estimation of normals, curvatures,and geodesics, as well as smoothing and curvature map com-putations, are performed on higher-resolution models.

To reduce search time further, we can trade deformation callsfor node expansion and pruning checks, as the latter opera-tions are less costly. Specifically, we use a parameter Lde f toindicate the first tree level at which mesh deformation is en-abled. Before this level, all nodes are assigned zero cost andonly tree pruning applies. Table 1 shows how different val-ues of Lde f can affect the number of deformation calls. Witha small Lde f , many deformation calls were made for corre-spondences which would end up being pruned. It is muchmore efficient to prune them without computing a deforma-tion, as is done by setting a higher Lde f . It is important tonote however that any solution returned by the search must

Table 1: Benefit of deferred deformation calls. Result is from

matching the wolves in Figure 2, with matching size N = 9.

Lde f = 6 Lde f = 7 Lde f = 8# Nodes expanded 2,803 4,332 4,926# Deformations 1,600 602 73

have an actual cost computed via mesh deformation. Afterall, tree pruning does not rate the correspondences. There-fore, if we seek matchings of size at least N, then we musthave Lde f < N. This ensures that all the returned solutionsare evaluated by our correspondence cost.

6. Results

We have tested our deformation-driven correspondence al-gorithm on various models with pronounced features. Theseconsist mainly of animals and human-like figures, which canvary significantly in pose, relative scale, part composition,and geometric detail. In the following experiments, we use10 geodesic bins and a neighborhood size that is 0.3 of themaximum pairwise geodesic distance to construct the curva-ture maps. As we are interested in coarse feature correspon-dence, we fix the feature count at 10. For automatic featureselection, we first set σ = 0.05 and γ = 0.2. If at least 10 fea-tures are found, we select the 10 most prominent ones (referto Section 4). Otherwise, we relax γ to 0.1 and repeat. Thisenables us to automatically select 10 features from each testmodel. Both tolerance values εsim and εgeod are set conserva-tively at 0.4. The matching size is determined automaticallyby examining the cost curve, as described in Section 5.3. Fora given matching size N, we set Lde f = N − 1 to minimizethe number of deformation calls.

We first take the five wolf models from the ISDB databaseand match one of them, shown in Figure 6(a), to the otherfour. No initial alignment is applied to any model pairs inour experiments, except for a uniform scale normalization.The final correspondence results are shown in Figure 6. Theydemonstrate the ability of our algorithm to handle pose; thereis only minor stretching and little change in the local ge-ometry of the models. In each case, 9 out of 10 featuresare properly matched and the one spurious feature is ig-nored by partial matching. We use the matching results tosmoothly morph between wolves in different poses (Fig-ure 7) by first utilizing the feature correspondences as in-put for dense cross-parameterization [KS04] and then inter-polate the corresponding Laplacian coordinates between themodels to obtain the intermediate geometries.

Moving on to more challenging examples, we match thetriceratops (top-left model in Figure 8) to a rhino, two pigswhich differ a great deal in the scale of their parts, and fi-nally the feline with wings to exemplify partial matching inthe presence of spurious parts. Figure 8 shows the automat-ically picked features on each model, the cost curves used

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H. Zhang & A. Sheffer & D. Cohen-Or & Q. Zhou & O. van Kaick & A. Tagliasacchi / Deformation-Driven Shape Correspondence

Figure 6: Correspondence results between the wolf given

in the box and four other wolves from the ISDB database,

each with 10 automatically extracted features. Again, blue

markers indicate unmatched features.

Figure 7: Morphing result on a pair of wolves, based on

dense cross-parameterization driven by initial correspon-

dences (Figure 6) computed by our algorithm.

to find the matching size, and the final correspondences re-turned. The cost curve starts at matching size 5 and stopswhen no solution is returned, as dictated by tree prunningbased on the threshold parameters chosen. As we can see,results from automatic correspondence may miss some fea-ture pairings which would have been detected by a human,e.g., all the tails of the animals. Nevertheless, these resultsare all correct partial matchings, which can be further re-fined by matching more features. This latter process wouldbe a more localized and thus efficient search.

To showcase another dense cross-parameterization exampleusing the computed feature correspondences as input, wecross-parameterized the triceratops and one of the pigs anduse the result to obtian a blended model, shown in Figure 9,where the front of the body is taken from the pig and the backfrom the triceratops. The same procedure was also applied tothe dino-skeleton and the raptor from Figure 1.

The ability of our algorithm to perform partial matching isevident in many of the visual examples shown so far: spuri-ous features or parts (e.g., see the triceratops vs. the feline),on one or both input models, are appropriately ignored inthe correspondence. Figure 10 gives another such example,where we note that if we switch the order of the two in-put meshes, the same correspondence is returned, since thepruning process and the correspondence cost are both sym-metric. Also, observe that the models we match all exhibitsymmetry. In no case has our algorithm produced a wrongcorrespondence resulting from symmetry switching.

The search for the best matching size is quite expensive com-putationally, since we maintain the same feature count, 10 in

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5 6 7 80.02

0.03

0.04

0.05

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Corr

espondence c

ost

5 6 70.04

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0.08

0.1

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Co

rre

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Figure 8: Automatic feature correspondence between the

triceratops (top left) and four different animals, as well as

itself. First column shows the animals with selected features.

Second column shows the correspondences (red markers are

matched features and blue markers are unmatched ones).

Cost curves shown in the last column are used to determine

the matching sizes (circled in the plots).

Figure 9: Blending the front of one animal with the back

of another created via dense cross-parameterization. Left: a

“prehistoric pig” (triceratops and pig in the second row of

Figure 8). Right: dino-skeleton and raptor from Figure 1.

our case, and need to perform tree search for small valuesof Lde f . The vertex counts for this series of models rangebetween 606 and 3,552; the triceratops has 2,832 vertices.It took from 20 minutes to more than one hour to performthese fully automatic correspondences. We observe that ma-jority of the search time was spent when Lde f ≤ 5. Therefore,it should be more efficient to construct the cost curve in re-verse order, i.e., computing matchings of larger sizes first.

c© 2008 The Author(s)Journal compilation c© 2008 The Eurographics Association and Blackwell Publishing Ltd.

H. Zhang & A. Sheffer & D. Cohen-Or & Q. Zhou & O. van Kaick & A. Tagliasacchi / Deformation-Driven Shape Correspondence

Figure 10: Partial matching with spurious object parts. A

woman with wings is matched with a dino-bird model.

Improving the speed of the solver, e.g., by setting a highertolerance for convergence, should also help. Another factorbehind the high search cost is the conservative tolerance val-ues we chose. A more stringent set of tolerances can providemany-fold speed-ups, but they are not always easy to findautomatically. One possible approach worth investigating isto start with low tolerances and then gradually increase themwhile observing the correspondence costs.

7. Discussions and limitations

Refinement: A number of factors influence the performanceof a correspondence algorithm. We first need a sufficient setof features to make the shapes matchable. However, the highcost of global search requires the feature count to be low,at least for the initial coarse correspondence. A low featurecount may lead to locally inaccurate results, e.g., when somefeatures are close to each other. We can then examine sev-eral candidate solutions (e.g., the top few from the our treesearch based on correspondence cost) and use refinementto provide a final ranking. The refinement procedure insertsmore features on two shapes and can simply match them upbased on geodesic distances to the initial correspondences.Meshes are deformed based on the refined correspondenceand the resulting costs provide the final ranking. Takingthis to the limit, we would arrive at cross-parameterization,e.g., [SAPH04].

Dependent feature picking: Shape extremities are not al-ways reliable as features for correspondence, e.g., considerthe unlikely case of matching to a part embedded in a largermodel where none of the model’s extremities lie on that part.A possible remedy is to make use of local shape signatures.For example, Gelfand et al. [GMGP05] extract feature pointsfrom one shape that have the most unusual signatures. Eachfeature is matched against all points on the second shape,where features are identified via thresholding the similaritybetween shape signatures. For this to work well, we needhigh-quality local shape signatures and as discussed in Sec-tion 3.1, this is a difficult problem in the non-rigid setting.

Figure 11: Matching the dino-skeleton with a brontosaurus.

The least-cost solution, shown on the right, is a wrong one.

Choosing tolerance values: It is generally difficult to se-lect the tolerance values automatically and conservative set-tings lead to high search cost. In practice, there are applica-tions, e.g., statistical modeling [ASK∗05], where feature cor-respondences are needed for a large set of models belongingto the same class. As the shape variations are not expectedto be too large, our algorithm should work efficiently undera relatively stringent tolerance setting.

Limitation of the deformation approach: While the de-formation view of shape correspondence is shown to be ef-fective in many cases, as demonstrated by the results so far,it cannot be expected to replicate the way humans performcorrespondence. Figure 11 shows the result of matching thedino-skeleton with a brontosaurus model via deformation.Indeed, it is the least-distorting way to correspond them, yetit is wrong. We argue that the reason we are able to matchthese two models correctly is that we can recognize them.Based purely on geometry, it can be an impossible task.

8. Conclusion and future work

It is generally believed that shape correspondence by humaninvolves an object recognition phase. That is, two shapesare first recognized as belonging to certain classes beforeidentifying feature correspondences. Without relying on thisphase, we develop a purely geometric approach which em-phasizes the global characteristics of shape correspondence.When large shape variations are tolerated, local shape de-scriptors can become unreliable. We take a deformation viewof shape correspondence and compute partial matchings be-tween two meshes which lead to low-distortion deformation.The ability of our algorithm to handle shape symmetry andshape variations in pose, local scale, part composition, andgeometric detail is demonstrated through various examples.

In future work, we would first like to improve the speed oftree search, while retaining an automatic algorithm. There-fore, we need effective means to find proper tolerance val-ues and techniques to prune the tree search more aggres-sively. Pruning and feature selection can both benefit fromhaving a high-quality local shape descriptor. The key ques-tion appears to be how to deal with local stretching or scalechanges near a feature point. We would like to look into theuse of feature-sensitive neighborhood traversals [LZH∗07]when constructing local shape descriptors.

c© 2008 The Author(s)Journal compilation c© 2008 The Eurographics Association and Blackwell Publishing Ltd.

H. Zhang & A. Sheffer & D. Cohen-Or & Q. Zhou & O. van Kaick & A. Tagliasacchi / Deformation-Driven Shape Correspondence

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